Over the past several decades there has been growing interest in the development of devices based on second-order nonlinear effects such as sum-frequency generation (SFG) and difference frequency generation (DFG) and devices based on third-order nonlinear effects such as Raman-resonant four-wave-mixing (FWM) and Kerr-induced four-wave-mixing (FWM). SFG, DFG and Kerr-induced FWM are parametric light-matter interactions that are not resonant with a material level and that are used in parametric converters and parametric amplifiers. Raman-resonant FWM is a light-matter interaction that is perfectly resonant or almost perfectly resonant with a characteristic energy level of the material such as a vibrational energy level and that is used in Raman converters. SFG, DFG and Kerr-induced FWM involve a pump radiation beam at frequency ωp, a signal radiation beam at frequency ωs, and an idler radiation beam at frequency ωi. Raman-resonant FWM involves a pump radiation beam at frequency ωp, a Stokes radiation beam at frequency ωstokes that is lower than the pump frequency, and an anti-Stokes radiation beam at frequency ωanti-stokes that is higher than the pump frequency. One also uses the terms signal and idler for the Stokes and anti-Stokes radiation beams, respectively, or vice versa, and uses ωs and ωi to denote their frequencies. Due to the wavelength versatility offered by SFG, DFG, Raman-resonant FWM and Kerr-induced FWM, these processes feature a multitude of application possibilities in different domains. One important example thereof are Raman converters, parametric converters and parametric amplifiers based on silicon which have attracted much attention because of their potential for application in optical communication systems.
Basically, Raman-resonant FWM and Kerr-induced FWM are interactions between two pump photons, one signal photon and one idler photon, and the frequencies of these photons ωp, ωs and ωi satisfy the relation 2ωp−ωs−ωi=0. For Raman-resonant FWM in silicon one has in addition that |ωp−ωs|=2π×15.6 THz, which is the Raman shift of silicon. In the case of SFG and DFG there is an interaction between 1 pump photon, one signal photon and one idler photon, and the frequencies of these photons ωp, ωs and ωo satisfy the relation ωp+ωs=ωi for SFG and ωp−ωs=ωi for DFG. The efficiency of all processes depends on the pump intensity and on the processes' phase mismatch. The linear part Δklinear of the phase mismatch for Raman-resonant FWM and Kerr-induced FWM is given byΔklinear=2kp−ks−ki where k{p,s,a}=ω{p,s,a}×n{p,s,a}/c are wave numbers with n{p,s,a} representing the effective indices of the pump, signal and idler waves, respectively. One can also write Δklinear as Δklinear=−β2(Δω)2−1/12β4(Δω)4 where β2=d2k/dω2 is the Group velocity dispersion (GVD) at the pump wavelength, β4=d4k/dω4 is the fourth-order dispersion at the pump wavelength, and Δω is the frequency difference between the pump and signal waves. For SFG the linear part Δklinear of the phase mismatch is given byΔklinear=kp+ks−kt 
For DFG the linear part Δklinear of the phase mismatch is given byΔklinear=kp−ks−ki 
The total phase mismatch for these processes also contains a nonlinear part that is function of the pump intensity, but since linear phase mismatches are considered here that are mostly much larger than the nonlinear part of the phase mismatch, the latter can be neglected in the remaining part of this text.
Due to their nonlinear nature, all above mentioned processes perform best at high optical intensities. These can be obtained by tightly confining the light for example in a nanowire waveguide and also by employing ring structures, whispering-gallery-mode disk resonators, or any other resonator structure in which the incoming light waves are resonantly enhanced. Regarding the requirement of having a small effective phase mismatch for the wavelength conversion processes, much progress has been made over the past several years. Regarding silicon-based converters, by engineering the dispersion of a silicon nanowire waveguide one can obtain phase-matched Kerr-induced FWM in the near-infrared for pump-signal frequency shifts with an upper limit of 52 THz (i.e., pump-signal wavelength differences up to 418 nm in the near-infrared region).
Notwithstanding the broad applicability of this phase-matched conversion technique, there are circumstances, applications, and materials where an alternative approach can be useful. First of all, not all materials used for SFG, DFG, Raman-resonant or Kerr-induced FWM are as easily workable as silicon to fabricate waveguide structures, which implies that not all materials can benefit from the waveguide-based phase-matching technique outlined above. Furthermore, even if one considers only a material such as silicon for which the waveguide-based phase-matching technique described above is well developed, it is important to know that, although the phase-matching bandwidth of the silicon nanowire referred to above is more than wide enough to enable phase-matched Raman-resonant FWM in the near-infrared at a pump-signal frequency shift of 15.6 THz, the dispersion-engineered geometry of the waveguide is such that one crucial advantage of using nanowires cannot be fully exploited. The particular advantage that cannot be fully exploited in that case is the possibility of having a fast recombination of two-photon-absorption-created free carriers at the waveguide boundaries. Indeed, the nanowire referred to above exhibits a relatively large free carrier lifetime (τeff≈3 ns), yielding substantial free carrier absorption (FCA) losses in the waveguide, and this decreases the wavelength conversion efficiency. Although these losses could be reduced by implementing around the nanowire carrier-extracting p-i-n diodes connected to a power supply, the advantage of using the low-cost intrinsic silicon-on-insulator platform would then be lost. Thus, in case one aims at keeping the fabrication cost and packaging cost as low as possible by opting for the intrinsic silicon-on-insulator platform, the challenge will be to enable efficient Raman-resonant FWM in the near-infrared wavelength domain using a nanowire that is not dispersion-engineered in a way that leads to substantial FCA losses.
Even if the increase in fabrication complexity and in cost when implementing carrier-extracting p-i-n diodes around the silicon nanowire as outlined above would not be considered as an issue, another challenge regarding silicon-based converters is in need of a solution. On one hand, the realization of phase-matched Kerr-induced FWM in silicon for pump-signal frequency shifts up to 52 THz in the near-infrared wavelength region fulfills the requirements of most wavelength conversion applications. On the other hand, specific applications in domains such as spectroscopy, sensing, industrial process control, environmental monitoring, biomedicine, and also telecommunications could benefit from wavelength conversion possibilities in both the near-infrared range and the mid-infrared region beyond 2 μm. These wavelength conversion possibilities should all ideally be available in one device, and preferably based on Kerr-induced FWM, which offers more wavelength flexibility than Raman-resonant FWM. It is extremely challenging, however, to engineer the dispersion of a silicon waveguide such that phase-matched Kerr-induced FWM is obtained for large pump-signal frequency shifts both in the near- and mid-infrared spectral regions. One approach to circumvent this problem of dispersion engineering would be to use the Kerr-induced FWM scheme based on two different pump frequencies, but the requirement of having a second pump frequency close to the desired idler frequency might be difficult to meet in the mid-infrared region, as widely tunable and practical mid-infrared pump sources are not so common yet. As such, for these specific applications the challenge will be to enable efficient, single-pump Kerr-induced FWM for a large pump-signal frequency shift in a spectral domain where the dispersion characteristics of the silicon nanowire are not optimally engineered for phase-matched Kerr-induced FWM. Also, if one aims at realizing a silicon-based source based on Kerr-induced FWM that, by changing just one parameter, can generate radiation at different wavelengths spread in the near- and mid-infrared spectral region, then one should use a technique different from phase matching. The development of such discretely-tunable sources would represent an important step in the search for low-cost, compact, room-temperature light sources tunable in the near- and mid-infrared. Such devices are still scarce nowadays but highly desirable for many applications, ranging from telecommunications and industrial process control, to environmental monitoring and biomedical analysis.
Also for SFG and DFG in silicon nanowires, there are dispersion engineering issues. To establish SFG and DFG in silicon, one usually applies strain on a silicon nanowire to induce the second-order nonlinearity that is needed for these processes and hence make the nanowire a quadratically nonlinear optical medium. Because of the very large pump-signal frequency shifts typically used in SFG and DFG, it is practically impossible to engineer the dispersion of the strained silicon nanowires in such a way that phase-matched SFG or DFG is obtained. Hence, it is challenging to achieve efficient SFG or DFG using only dispersion engineering.
One suggestion has been to establish quasi-phase-matching for SFG, DFG, Raman-resonant FWM or Kerr-induced FWM using heterogeneous materials. This traditional quasi-phase-matching technique for these nonlinear processes can be understood as follows: In case nothing is done about the phase mismatch, the idler intensity for radiation would continuously oscillate along the propagation path between a maximal value and zero, as the phase-mismatch-induced dephasing of the fields—this dephasing evolves periodically with the propagation distance—causes the nonlinear optical processes to either increase or decrease the idler intensity along the propagation path. When using traditional quasi-phase-matching for these processes, one adjusts the propagation regions behind the positions of maximal idler intensity, so that one does not have a total drop down of the idler intensity in these regions but at the same time the fields' dephasing, accumulated up to the positions of maximal idler intensity, can evolve back to zero in these adjusted regions. Hence, after traversing these adjusted areas the idler intensity can start growing again towards a maximum. The type of “adjustment” that needs to be applied to these propagation regions is that the susceptibility should be made zero there for the Raman-resonant or Kerr-induced FWM processes or reversed in sign for the SFG and DFG processes, so that these nonlinear processes cannot establish a decrease of the idler intensity in these areas whereas the fields' dephasing can still evolve back to zero. This is done using a heterogeneous conversion medium of which the characteristics are periodically manipulated. This is a complex approach and disadvantageous from a practical point of view.